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The paper addresses the problem of identification of nonlinear characteristics in a certain class of discrete-time block-oriented systems. The systems are driven by random stationary white processes (independent and identically distributed input sequences) and disturbed by stationary, white, or colored random noise. The prior information about nonlinear characteristics is nonparametric. In order to construct identification algorithms, the orthogonal wavelets of compact support are applied, and a class of wavelet-based models is introduced and examined. It is shown that under moderate assumptions, the proposed models converge almost everywhere (in probability) to the identified nonlinear characteristics, irrespective of the noise model. The rule for optimum model-size selection is given and the asymptotic rate of convergence of the model error is established. It is demonstrated that, in some circumstances, the wavelet models are, in particular, superior to classical trigonometric and Hermite orthogonal series models worked out earlier.